Simplifying Complex Numbers Absolute Values Explained

In the realm of complex numbers, the absolute value, often referred to as the modulus, represents the distance of a complex number from the origin in the complex plane. Understanding how to simplify expressions involving absolute values of complex numbers is a fundamental skill in complex analysis. This comprehensive guide will delve into the process of writing expressions without absolute value symbols, providing a step-by-step approach coupled with illustrative examples. We will focus on simplifying the expression $\left|\frac{10i}{i-3}\right|$, demonstrating the techniques required to achieve a complete solution.

Understanding the Absolute Value of Complex Numbers

Before diving into the simplification process, it's crucial to grasp the concept of the absolute value of a complex number. A complex number, typically denoted as $z$, can be expressed in the form $z = a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, defined as $i^2 = -1$. The absolute value (or modulus) of $z$, written as $|z|$, is calculated using the formula:

$|z| = \sqrt{a^2 + b^2}$

This formula stems from the Pythagorean theorem, where $a$ and $b$ represent the lengths of the legs of a right triangle, and $|z|$ represents the length of the hypotenuse, which is the distance from the origin to the point $(a, b)$ in the complex plane. Understanding this geometrical interpretation is key to visualizing and manipulating complex numbers effectively. When dealing with fractions of complex numbers within absolute value symbols, we can leverage a critical property: the absolute value of a quotient is the quotient of the absolute values. Mathematically, this is expressed as:

$\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}$

where $z_1$ and $z_2$ are complex numbers, and $z_2$ is not equal to zero. This property allows us to break down complex expressions into simpler components, making the simplification process more manageable. In our specific problem, $\left|\frac{10i}{i-3}\right|$, we can apply this property to separate the absolute value of the numerator and the absolute value of the denominator, paving the way for further simplification. Furthermore, when working with absolute values, it is often beneficial to eliminate complex numbers from the denominator. This is achieved by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number $a + bi$ is $a - bi$, and multiplying a complex number by its conjugate results in a real number, specifically $a^2 + b^2$. This technique is crucial for simplifying expressions and ultimately removing the absolute value symbols, as it transforms the denominator into a real number, making the calculation of the absolute value straightforward. The interplay between these concepts – the definition of absolute value, the quotient property, and the use of conjugates – forms the foundation for effectively simplifying complex number expressions.

Step-by-Step Simplification of $\left|\frac{10i}{i-3}\right|$

Let's now apply these principles to simplify the given expression, $\left|\frac{10i}{i-3}\right|$. Our goal is to eliminate the absolute value symbols and express the result in its simplest form. The process involves several key steps, each building upon the previous one to progressively simplify the expression. We start by applying the property that the absolute value of a quotient is the quotient of the absolute values. This allows us to separate the expression into two simpler absolute values, one for the numerator and one for the denominator. This initial step is crucial as it breaks down the complex expression into more manageable parts, making it easier to apply the definition of absolute value. Then, we focus on simplifying the absolute value of the denominator, which involves a complex number. To do this, we employ the technique of multiplying both the numerator and the denominator by the conjugate of the denominator. This is a standard method for rationalizing complex denominators and eliminating imaginary parts from the denominator. The conjugate of $i - 3$ is $-i - 3$, and multiplying both the numerator and denominator by this conjugate will transform the denominator into a real number. This step is critical because it allows us to apply the definition of absolute value more easily, as the absolute value of a real number is simply its magnitude. After multiplying by the conjugate, we simplify the resulting expression by performing the necessary multiplications and combining like terms. This step involves careful algebraic manipulation to ensure accuracy and to prepare the expression for the final application of the absolute value definition. Finally, we apply the definition of absolute value to both the numerator and the simplified denominator. This involves calculating the square root of the sum of the squares of the real and imaginary parts for each complex number. Once we have the absolute values calculated, we can simplify the fraction to obtain the final result, which is the expression without absolute value symbols. Each of these steps is carefully designed to break down the complex problem into smaller, more manageable tasks, ultimately leading to a complete and simplified solution. By understanding and mastering these steps, one can confidently tackle a wide range of problems involving absolute values of complex numbers.

1. Apply the Quotient Property of Absolute Values

The first step in simplifying $\left|\frac{10i}{i-3}\right|$ is to use the property that the absolute value of a quotient is the quotient of the absolute values. This allows us to rewrite the expression as:

$\left|\frac{10i}{i-3}\right| = \frac{|10i|}{|i-3|}$

This separation makes the expression easier to handle, as we can now focus on simplifying the absolute value of the numerator and the denominator separately. This step is a fundamental application of a core property of absolute values and sets the stage for the subsequent simplification steps. The separation allows us to treat each part of the expression individually, applying specific techniques to simplify each absolute value. This is a common strategy in mathematics: breaking down a complex problem into smaller, more manageable parts. By isolating the numerator and denominator, we can more easily apply the definition of absolute value and use algebraic manipulations to arrive at the final simplified expression.

2. Calculate the Absolute Value of the Numerator

The numerator is $10i$, which can be written in the form $a + bi$ as $0 + 10i$. The absolute value of $10i$ is calculated as:

$|10i| = \sqrt{0^2 + 10^2} = \sqrt{100} = 10$

This step involves applying the definition of absolute value directly. The absolute value of a complex number is its distance from the origin in the complex plane, which is calculated using the Pythagorean theorem. In this case, the complex number $10i$ lies on the imaginary axis at a distance of 10 units from the origin. Therefore, its absolute value is 10. This calculation is straightforward and provides a numerical value for the numerator's absolute value, which we will use in the next steps to further simplify the overall expression. By calculating the absolute value of the numerator, we have reduced one part of the problem to a simple number, making the subsequent steps more manageable. This is a key aspect of problem-solving: breaking down a complex problem into smaller, easier-to-solve parts.

3. Calculate the Absolute Value of the Denominator

The denominator is $i - 3$, which can be written as $-3 + i$. The absolute value of $i - 3$ is calculated as:

$|i - 3| = \sqrt{(-3)^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10}$

Similar to the numerator, we apply the definition of absolute value to the complex number in the denominator. The complex number $i - 3$ has a real part of -3 and an imaginary part of 1. We calculate its absolute value by taking the square root of the sum of the squares of these parts. This results in $\sqrt{10}$, which is the distance of the point (-3, 1) from the origin in the complex plane. This calculation is crucial for simplifying the denominator, as it replaces the complex number within the absolute value symbols with a real number. This allows us to combine the results from the numerator and denominator to obtain the final simplified expression. By calculating the absolute value of the denominator, we have transformed it into a real number, which is essential for the subsequent simplification steps.

4. Substitute the Absolute Values Back into the Expression

Now we substitute the calculated absolute values back into the expression:

$\frac{|10i|}{|i-3|} = \frac{10}{\sqrt{10}}$

This step combines the results from the previous steps, bringing together the simplified numerator and denominator. We have replaced the absolute value expressions with their numerical values, which allows us to focus on simplifying the resulting fraction. This substitution is a crucial step in the overall simplification process, as it allows us to move from dealing with absolute values of complex numbers to working with a simple fraction involving real numbers. This transition makes the final simplification much more straightforward. By substituting the calculated absolute values, we have created a fraction that can be easily simplified using basic algebraic techniques.

5. Simplify the Fraction

To simplify the fraction, we rationalize the denominator by multiplying both the numerator and the denominator by $\sqrt{10}$:

$\frac{10}{\sqrt{10}} = \frac{10 \times \sqrt{10}}{\sqrt{10} \times \sqrt{10}} = \frac{10\sqrt{10}}{10} = \sqrt{10}$

This step involves a standard technique for simplifying fractions with radicals in the denominator. Rationalizing the denominator means eliminating the radical from the denominator, which is often preferred in mathematical expressions. We achieve this by multiplying both the numerator and denominator by the radical in the denominator, which in this case is $\sqrt{10}$. This multiplication results in a perfect square in the denominator, allowing us to simplify the fraction further. The simplification process involves canceling common factors, in this case, the factor of 10. The final result is $\sqrt{10}$, which is the simplified expression without absolute value symbols. This step is the culmination of the simplification process, where we transform the fraction into its simplest form, providing a clear and concise answer. By rationalizing the denominator, we have obtained an expression that is considered mathematically elegant and easy to interpret.

Final Answer

Therefore, $\left|\frac{10i}{i-3}\right|$ simplified completely is $\sqrt{10}$. This result is obtained by systematically applying the properties of absolute values, calculating the absolute values of the numerator and denominator separately, and simplifying the resulting fraction. The step-by-step approach ensures accuracy and clarity in the solution process. Understanding these techniques is crucial for handling more complex expressions involving absolute values of complex numbers.

To accurately address the task of rewriting the expression without absolute value symbols, it is essential to correctly apply the mathematical definitions and properties of absolute values in the context of complex numbers. The initial interpretation and breakdown of the expression dictate the subsequent steps. Misunderstanding these foundational concepts can lead to significant deviations from the correct solution path, resulting in an incorrect final answer. For example, failing to apply the quotient rule of absolute values properly, or incorrectly calculating the magnitude of a complex number, can compound errors throughout the simplification process. Attention to detail is equally critical. Complex number calculations, especially those involving square roots and fractions, are prone to errors if meticulous care is not taken in each step. A simple arithmetic mistake in squaring a number or in rationalizing the denominator can alter the outcome. It is vital to double-check each calculation and ensure that all algebraic manipulations are performed correctly. This includes correctly applying the distributive property, simplifying radicals, and combining like terms. Furthermore, a clear and organized approach to problem-solving is crucial for minimizing errors. Writing down each step neatly and logically helps in keeping track of the operations performed and reduces the likelihood of overlooking a mistake. It is advisable to break down the problem into smaller, manageable parts and tackle each part systematically. This methodical approach not only makes the problem less daunting but also allows for easier error detection and correction. Regularly reviewing the steps and verifying the intermediate results can help ensure accuracy. In summary, accuracy and precision in the simplification of mathematical expressions, especially those involving complex numbers and absolute values, are paramount. A solid grasp of the underlying mathematical principles, coupled with meticulous attention to detail and a systematic approach, forms the basis for successfully navigating these problems and arriving at the correct solution.

In conclusion, simplifying expressions involving absolute values of complex numbers requires a solid understanding of the definition of absolute value, the properties of complex numbers, and careful algebraic manipulation. By following a systematic approach, such as the one outlined in this guide, one can confidently tackle these problems and arrive at the correct solution. The key steps include applying the quotient property of absolute values, calculating the absolute values of the numerator and denominator separately, and simplifying the resulting fraction. Practice and attention to detail are essential for mastering these techniques and ensuring accuracy in your calculations. The ability to simplify complex expressions is a fundamental skill in mathematics, with applications in various fields such as physics, engineering, and computer science. Mastering these techniques not only enhances your mathematical proficiency but also provides a valuable tool for problem-solving in a wide range of contexts. The journey through simplifying complex expressions is not just about finding the right answer; it's about developing a deeper understanding of mathematical concepts and honing your problem-solving skills. Each problem is an opportunity to learn, to refine your techniques, and to build confidence in your ability to tackle mathematical challenges. So, embrace the challenge, practice diligently, and enjoy the process of unraveling the intricacies of complex numbers and their absolute values.